criterion Resolution of an optical system http micromagnetfsuedu primer The resolving power of an optical system is limited by the diffraction occurring at the optical path every time there is an ID: 1031169
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1. 1High Resolution TEM
2. Rayleigh criterion Resolution of an optical systemhttp://micro.magnet.fsu.edu/primerThe resolving power of an optical system is limited by the diffraction occurring at the optical path every time there is an aperture/diaphragm/lens.The aperture causes interference of the radiation (the path difference between the green waves results in destructive interference while the path difference between the red waves results in constructive interference). An object such as point will be imaged as a disk surrounded by rings. The image of a point source is called the Point Spread Function1 point 2 points unresolved2 pointsresolved Point spread function (real space)
3. Tube lensBack focal plane apertureIntermediate image planeSampleObjectiveDiffraction spoton image plane= Point Spread FunctionAperture and resolution of an optical system3
4. Tube lensBack focal plane apertureIntermediate image planeSampleObjectiveDiffraction spoton image plane= Point Spread FunctionAperture and resolution of an optical system4
5. Tube lensBack focal plane apertureIntermediate image planeSampleObjectiveDiffraction spoton image plane= Point Spread FunctionAperture and resolution of an optical system5
6. Aperture and resolution of an optical systemThe larger the aperture at the back focal plane (diffraction plane), the larger and higher the resolution (smaller disc in image plane)SampleObjectiveTube lensBack focal plane apertureIntermediate image planeNA = n sin() = light gathering anglen = refractive index of samplewhere:Diffraction spoton image plane= Point Spread Function6
7. New concept: Contrast Transfer Function (CTF)7
8. Optical Transfer Function (OTF)ObjectObservedimage(Spatial frequency,periods/meter)K or gOTF(k)1ImagecontrastResolution limitKurt Thorn, University of California, San Francisco8
9. Definitions of Resolution|k|OTF(k)1As the OTF cutoff frequencyAs the Full Width at Half Max(FWHM) of the PSFAs the diameter of the Airy disk(first dark ring of the PSF)= “Rayleigh criterion”Kurt Thorn, University of California, San FranciscoAiry disk diameter≈ 0.61 /NAFWHM≈ 0.353 /NA1/kmax= 0.5 /NA9
10. Resolution CriteriaRayleigh’sdescriptionAbbe’sdescription0.6l/NAl/2NAAberration free systemsKurt Thorn, University of California, San Francisco10
11. Remember: images can be considered sums of wavesanother waveone wave(2 waves)+=(10000 waves)+ (…) =… or “spatial frequency components”(25 waves)+ (…) =Kurt Thorn, University of California, San Francisco11
12. Remember: reciprocal/frequency spaceFrequency (how many periods/meter?)DirectionAmplitude (how strong is it?)Phase (where are the peaks & troughs?)perioddirectionTo describe a wave, specify:kykxDistance from originDirection from originMagnitude of value at the pointPhase of numberA wave can also be describedby a complex number at a point:complexk = (kx , ky)Kurt Thorn, University of California, San Francisco12
13. kykxRemember: frequency spacekykxand the Fourier transformFourierTransform13
14. ObservableRegionkykxThe Transfer Function Lives in Frequency SpaceObject|k|OTF(k)ObservedimageKurt Thorn, University of California, San Francisco14
15. Remember: the Properties of the Fourier TransformSymmetry:The Fourier Transform of the Fourier Transform is the original imageCompleteness:The Fourier Transform contains all the information of the original imageFouriertransformKurt Thorn, University of California, San Francisco15
16. The OTF and ImagingFourierTransformTrueObjectObservedImageOTF==?convolutionPSFKurt Thorn, University of California, San Francisco16
17. Convolutions(f g)(r) = f(a) g(r-a) daWhy do we care?They are everywhere…The convolution theorem:Ifthenh(r) = (fg)(r),h(k) = f(k) g(k)A convolution in real space becomesa product in frequency space & vice versaSo what is a convolution, intuitively?“Blurring”“Drag and stamp”=fgfg=xxyxyySymmetry: g f = f gKurt Thorn, University of California, San Francisco17
18. Resolution in HRTEMIn optical microscopy, it is possible to define point resolution as the ability to resolve individual point objects. This resolution can be expressed (using the criterion of Rayleigh) as a quantity independent of the nature of the object. The resolution of an electron microscope is more complex. Image "resolution" is a measure of the spatial frequencies transferred from the image amplitude spectrum (exit-surface wave-function) into the image intensity spectrum (the Fourier transform of the image intensity). This transfer is affected by several factors:the phases of the diffracted beams exiting the sample surface, additional phase changes imposed by the objective lens defocus and spherical aberration, the physical objective aperture, coherence effects that can be characterized by the microscope spread-of-focus and incident beam convergence.For thicker crystals, the frequency-damping action of the coherence effects is complex but for a thin crystal, i.e., one behaving as a weak-phase object (WPO), the damping action can best be described by quasi-coherent imaging theory in terms of envelope functions imposed on the usual phase-contrast transfer function.The concept of HRTEM resolution is only meaningful for thin objects and, furthermore, one has to distinguish between point resolution and information limit.O'Keefe, M.A., Ultramicroscopy, 47 (1992) 282-29718
19. 19Microscope: Transforms each point on the specimen into an extended region (at best, a circular disk) in the final image.Each point on the specimen may be different, we describe the specimen by a specimen function, f(x,y).The extended region in the image which corresponds to the point (x,y) in the specimen is then described as g(x,y)
20. 20Two nearby points, A and B, they will produce two overlapping images gA and gBEach point in the image has contributions from many points in the specimen.How much each point in the specimen contributes to each point in the image
21. 21h(r): How information in real space is transferred from the specimen to the imageH(u): How information (or contrast) in u space is transferred to the image.H(u) is the contrast transfer function.Now these three Fourier transforms are related byG(u) = H(u) F(u)So a convolution in real space gives multiplication in reciprocal space The factors contributing to H(u) include:Apertures The aperture function A(u)Attenuation of the wave The envelope function E(u)Aberration of the lens The aberration function B(u)H(u) is the Fourier transform of h(r)
22. 22The aperture function A(u): The objective diaphragm cuts off all values of u (spatial frequencies) greater than (higher than) some selected value governed by the radius of the aperture.The envelope function E(u):Has the same effect but is a property of the lens itself, and so may be either more or less restricting than A(u). The aberration function B(u):Is usually expressed asWe write H(u) as the product of these three termsH(u) = A(u) E(u) B(u)Defocus of the objective lensElectron wavelengthSpherical aberration coefficient
23. High spatial frequencies correspond to large distances from the optic axis in the DP. The rays which pass through the lens at these large distances are bent through a larger angle by the objective lens. They are not focused at the same point by the lens, because of spherical aberration, and thus cause a spreading of the point in the image. The result is that the objective lens magnifies the image but confuses the fine detail. The resolution we require in HRTEM is limited by this ‘confusion’Each point in the specimen plane is transformed into an extended region (or disk) in the final image.Each point in the final image has contributions from many points in the specimen.
24. 24Spherical aberrationElectrons with high spatial frequency (large distance from optical axis) are bent through a larger angleCauses: Spreading of the point in the imageThe resolution we require in HRTEM is limited by this ‘confusion’Each point in the specimen plane is transformed into an extended region (or disk) in the final image.Each point in the final image has contributions from many points in the specimen.
25. 25Weak Phase-Object ApproximationFor a very thin specimen, the amplitude of a transmitted wave function will be linearly related to the projected potential of the specimen.The projected potential is taking account of variations in the z-direction, and is thus very different for an electron passing through the center of an atom compared to one passing through its outer regions.WPOA fails for an electron wave passing through the center of a single uranium atom! An atomic layer of U would be too thick for the WPOA.A model to represent the specimen: Assumptions:Sample very thinAmplitude A(x,y) = 1 (unity)Represent the specimen as phase objectSmall absorptionsVt(x,y) <<1
26. 26Intensity transfer function: T(u) = A(u) E(u) 2sin χ(u)(Objective lens transfer function)T(u) = A(u) exp (iχ) exp(-π2Δ2λ2u4/2) exp(-π2uc2q2)Where: q = Cs λ3u3 + Δf λ uΔ H(u) = A(u) E(u) B(u) = A(u) E(u) exp(iχ(u))Only the imaginary part
27. 27Transfer function T(u)1. T(u), formulation applies to any specimen2. T(u) is not the CTF of HRTEMThe image wave function is not an observable quantity! What we observe in an image is contrast, or the equivalent in optical density, current readout, etc., and this is not linearly related to the object wave function. Fortunately, there is a linear relation involving observable quantities under the special circumstances, where the specimen acts as a WPOIf we have WPO : T(u) called CTFNo amplitude contribution Output of the transmission system is an observable quantity (image contrast)
28. 28T(u) = A(u) E(u) 2sin χ(u)Ignore E(u)T(u) = 2 A(u) sin χ(u)Phase distortion functionhas the form of a phase shift expressed as 2π/λ times the path difference traveled by those waves affected by spherical aberration (Cs), defocus (Δz), and astigmatism (Ca).CTF is: OscillatoryThere are bands of good transmission separated by gaps (zeros) where no transmission occursMaxima: when phase-distortion function assumes multiple odd values of ±π/2Zero contrast for χ(u) = multiple of ±π
29. 29T(u)<0 Positive phase contrast: Phase shift of –π/2 due to diffraction Atoms appear dark against bright backgroundT(u)>0 Negative phase contrast: Phase shift of +π/2 due to diffraction (adds amplitude: 'in phase') Atoms appear bright against dark backgroundT(u)=0 No detail in the image (assuming Cs>0)
30. Contrast transfer functionksin χ(k) = sin(πλ∆fk2 +1/2πCsλ3k4) 30
31. Contrast transfer functionk:parameters: λ=0.0025 nm (200 kV), cs =1.1 mm, Δf=60 nmsin χ(k) = sin(πλ∆fk2 +1/2πCsλ3k4) sin χ(k) The CTF oscillates between -1 (negative contrast transfer) and +1 (positive contrast transfer). The exact locations of the zero crossings (where no contrast is transferred, and information is lost) depends on the defocus.31
32. Point resolutionPoint resolution: related to the finest detail that can be directly interpreted in terms of the specimen structure. Since the CTF depends very sensitively on defocus, and in general shows an oscillatory behavior as a function of k, the contribution of the different scattered beams to the amplitude modulation varies. However, for particular underfocus settings the instrument approaches a perfect phase contrast microscope for a range of k before the first crossover, where the CTF remains at values close to –1. It can then be considered that, to a first approximation, all the beams before the first crossover contribute to the contrast with the same weight, and cause image details that are directly interpretable in terms of the projected potential. Optimisation of this behaviour through the balance of the effects of spherical aberration vs. defocus leads to the generally accepted optimum defocus1 −1.2(Csλ)1/2. Designating an optimum resolution involves a certain degree of arbitrariness. However, the point where the CTF at optimum defocus reaches the value –0.7 for k = 1.49C−1/ 4λ−3/4 is usually taken to give the optimum (point) resolution (0.67C1/4λ3/4). This means that the considered passband extends over the spatial frequency region within which transfer is greater than 70%. Beams with k larger than the first crossover are still linearly imaged, but with reverse contrast. Images formed by beams transferred with opposite phases cannot be intuitively interpreted. 32
33. The Contrast Transfer FunctionThe effect of different Cs and Δf on CTFPlot: T(u) = sin χ
34. Important points to notice:CTF is oscillatory: there are "passbands" where it is NOT equal to zero (good "transmittance") and there are "gaps" where it IS equal (or very close to) zero (no "transmittance").When it is negative, positive phase contrast occurs, meaning that atoms will appear dark on a bright background.When it is positive, negative phase contrast occurs, meaning that atoms will appear bright on a dark background.When it is equal to zero, there is no contrast (information transfer) for this spatial frequency.
35. Other important features:CTF starts at 0 and decreases, thenCTF stays almost constant and close to -1 (providing a broad band of good transmittance), thenCTF starts to increase, andCTF crosses the u-axis, and thenCTF repeatedly crosses the u-axis as u increases.CTF can continue forever but, in reality, it is modified by envelope functions and eventually dies off. Effect of the envelope functions can be represented as
36. Scherzer defocusEvery zero-crossing of the graph corresponds to a contrast inversion in the image. Up to the first zero-crossing k0 the contrast does not change its sign. The reciprocal value 1/k0 is called Point Resolution.The defocus value which maximizes this point resolution is called the Scherzer defocus. Optimum defocus: At Scherzer defocus, one aims to counter the term in u4 with the parabolic term Δfu2 of χ(u). Thus by choosing the right defocus value Δf one flattens χ(u) and creates a wide band where low spatial frequencies k are transferred into image intensity with a similar phase.Working at Scherzer defocus ensures the transmission of a broad band of spatial frequencies with constant contrast and allows an unambiguous interpretation of the image.36
37. 37DefocusThe presence of zeros in CTF means that we have gaps in the output signal. Obviously, the best CTF is the one with the fewest zeros and with the broadest band of good transmittance (where CTF is close to -1). Back in 1949 Scherzer suggested an optimum defocus condition which occurs at: this value is now called "1 scherzer".Sometimes the value of 1.2 scherzer is called "Scherzer defocus" (we are going to call the value of 1.2 scherzer as "extended scherzer" throughout this manual). "Extended scherzer" provides even broader band of transmittance with CTF still close enough to -1.
38. http://www.maxsidorov.com/ctfexplorer/webhelp/effect_of_defocus.htmΔ f = - (Csλ)1/2Δ f = -1.2(Csλ)1/2Scherzer conditionExtended Scherzer condition38Scherzer defocus
39. 39Optimum focus conditions:Scherzer focus:Scherzer resolution:Extended Scherzer focus:Extended Scherzer resolution: (Point resolution) Optimum aperture size:2) Optimum information limit: Instrumental resolution given by the envelope function b/d
40. The Envelope functionTeff = T(u)EcEaThe resolution is also limited by the spatial coherence of the source and by chromatic effect:The envelope function imposes a “virtual aperture” in the back focal plane of the objective lens
41. Information limit goes well beyond point resolution limit for FEG microscopes (due to high spatial and temporal coherency).For the microscopes with thermionic electron sources (LaB6 and W), the info limit usually coincides with the point resolution.Phase contrast images are directly interpretable only up to the point resolution (Scherzer resolution limit).If the information limit is beyond the point resolution limit, one needs to use image simulation software to interpret any detail beyond point resolution limit.
42. Information limitInformation limit: corresponds to the highest spatial frequency still appreciably transmitted to the intensity spectrum. This resolution is related to the finest detail that can actually be seen in the image (which however is only interpretable using image simulation). For a thin specimen, such limit is determined by the cut-off of the transfer function due to spread of focus and beam convergence (usually taken at 1/e2 or at zero). These damping effects are represented by ED or Etc a temporal coherency envelope (caused by chromatic aberrations, focal and energy spread, instabilities in the high tension and objective lens current), and Ea or Esc is the spatial coherency envelope (caused by the finite incident beam convergence, i.e., the beam is not fully parallel).The Information limit goes well beyond point resolution limit for FEG microscopes (due to high spatial and temporal coherency). For the microscopes with thermionic electron sources (LaB6 and W), the info limit usually coincides with the point resolution. \The use of FEG sources minimises the loss of spatial coherence. This helps to increase the information limit resolution in the case of lower voltage ( ≤ 200 kV) instruments, because in these cases the temporal coherence does not usually play a critical role. However the point resolution is relatively poor due to the oscillatory behavior of the CTF. On the other hand, with higher voltage instruments, due to the increased brightness of the source, the damping effects are always dominated by the spread of focus and FEG sources do not contribute to an increased information limit resolution. 42
43. Damped contrast transfer functionMicroscope examples PointresolutionInformationlimitSpatial envelopeTemporalenvelope(Scherzer)Thermoionic, 400 kVFEG, 200 kV43
44. CTF is oscillatory: there are "passbands" where it is NOT equal to zero (good "transmittance") and there are "gaps" where it IS equal (or very close to) zero (no "transmittance").When it is negative, positive phase contrast occurs, meaning that atoms will appear dark on a bright background.When it is positive, negative phase contrast occurs, meaning that atoms will appear bright on a dark background.When it is equal to zero, there is no contrast (information transfer) for this spatial frequency.At Scherzer defocus CTF starts at 0 and decreases, thenCTF stays almost constant and close to -1 (providing a broad band of good transmittance), thenCTF starts to increase, andCTF crosses the u-axis, and thenCTF repeatedly crosses the u-axis as u increases.CTF can continue forever but, in reality, it is modified by envelope functions and eventually dies off. Important points to notice44
45. 45Defocus: 1 scherzer ("True" Scherzer defocusDefocus: 1.2 scherzer ("Extended" Scherzer defocus). In general, this is the best defocus to take HR-TEM images.
46. 46Defocus: 1.9 scherzer ("2nd Passband" defocus). Produces a nice and broad passband which starts NOT at zero. CTF is positive so that it produces a negative phase contrast ("white atoms")Defocus: 0.4 scherzer ("Minimum Contrast" defocus)
47. 47Defocus: 0 (low contrast, not good). Note: Zero-defocus images are NOT "Minimum Contrast" images. Minimum contrast occurs at about 0.4 scherzer (shown above).Defocus: 10 scherzer (large and generally not good for HR-TEM)
48. 48